In conceptual chemistry, the aromaticity notion stands as one of the main pillars of structural understanding of molecular stability and reactivity; it spans all relevant periods of modern chemistry, suffering continuous revivals as the development of the physico-mathematical methods allows [1–5].

Historically, the custom definition of aromaticity is that it characterizes the planar molecules with (4n + 2) π-electrons [6], favoring substitution while resisting addition reactions [7,8], corresponds with high stability respecting the anti-aromatic structure [9], and ultimately correlates with low diamagnetism or magnetic susceptibility [10,11]. However, through the discovery of the non-planar feature of the most aromatic structure—benzene [12], along the generalizations of the Hückel rule for conjugated systems leading with the conjugated circuits [13,14], topological conjugated structures [15–17], up to the aromatic zones of molecular fragments [18,19], the aromaticity concept appears as a quite versatile concept that still needs proper quantification.

The main difficulty in capping the observability character of aromaticity may reside in the fact it does not directly relate to the ground state energy of molecules, but rather with their excited and valence orbitals–a condition closely related with the π electrons delocalized in the structure. Such aromaticity paradox of quantifying molecular stability (that usually is conducted by a Variational algorithm towards the ground state) by means of frontier pi electrons, may be considered as the main origin for the most inconsistencies and controversial points relating to the aromaticity concept in general and of its scales’ realization in particular.

Still, advancing aromaticity scales that employ certain physico-chemical molecular properties are useful at least for understanding whether certain molecular properties are inter-related and whether they enter or not the aromaticity sphere of definition. For that, the aromaticity scales should be always judged and validated in a comparative manner through concluding merely qualitatively upon the benchmarking degree of inter-correlation. This way, the aromaticity concept works best within the “transitivity thinking”: if the scale A₁ correlates with the scales A₂ and A₃ with the same degree then the properties underlying the last two scales should be as well correlated and they may be regarded as different faces of the same molecular property. In short, the aromaticity concept and especially through its scales has the role of ordering among the molecular properties in general and of reactivity indices in special.

Returning to the aromaticity definition, it seems that the two main routes for its quantitative evaluation are the energetic and geometric ways; although within a Variational approach they should be closely related, i.e., minimum global energy corresponds with the geometry optimization, since the aromaticity paradox described above the two sides of molecular structures open two different approaches for introducing quantitative indices of aromaticity. For instance, based on geometric (also extended to topological) criteria, the consecrated harmonic oscillator based-molecular aromaticity (HOMA) [20,21] and the recent topological paths and aromatic zones (TOPAZ) [18], as well as the ultimate topological index of reactivity (TIR) [16] indices describe in various extent the influence the nuclei motion, the molecular fragment conjugation, or the site with the maximum probability (entropy) in electrophilic substitution have on aromaticity viewed as increasing (for the first two indices) or decreasing (for the last index) as the more delocalized π-electrons in question, respectively. On the other hand, from the energetic perspective, the resonance energy (RE) or its version reported per concerned π-electrons (REPE) [22–24], together with the heat (enthalpy) of formation (as a thermodynamic stabilization criterion of energy) [25], give another alternative to quantify aromaticity rises with their increase (for the first two indices) and decrease (for the last index), respectively. Moreover, other groups of methods in aromaticity evaluation are developed by employing the molecular magnetic [17,26–30] as well as based on electron delocalization [31–37] properties. It is therefore clear that the aromaticity scale is neither unique in trend nor in quantification and deserves further geometrical-energetic comparative investigation.

In this context, wishing to provide a fresh geometrical vis-à-vis of energetic aromaticity discussion, the present paper introduces the atoms-in-molecules compactness form of aromaticity that is then specialized both at geometrical level though the polarizability information and within the energetic framework though the electronegativity and chemical hardness reactivity indices. Following this, they will be used for ordering ten basic organic compounds, aromatic annulens, amines, hydroxyarenes, and heterocycles with nitrogen, against the corresponding aromaticity within most common semi-empirical and ab initio quantum chemical methods. The present considerations and results aim to further clarify the relationship between the electronegativity and chemical hardness in modeling the molecular stability/reactivity/aromaticity, as well the “computational distance” among their output furnished by various quantum mechanical schemes used in structural chemistry. For all these, the aromaticity is involved both as the motif and the tool having overall the manifestly unifying character among the fundamental concepts and computational schemes of chemistry.

The above reactivity indices-based aromaticity scales are now computed within the presented quantum chemical schemes for a limited yet significant series of benzenoids (see Table 2) containing the “life” atoms of Table 1. The atoms-in-molecule of aromaticity scales of polarizability, electronegativity and chemical hardness of Equations (12), (17), and (20) are directly computed upon the formulas given by Equations (8), (13), and (18), respectively; they are based exclusively on the data of Table 1 with the AIM results listed in Table 2, the 5th, 9th, and 10th columns, respectively.

For the post-bonding evaluations of the same indices, one must note the special case of polarizability that is computed upon the general Equation (11)–thus involving the molecular volume pre-computation. Here it is worth commenting on the fact that one can directly compute the molecular polarizability in various quantum schemes–however, with the deficiency that such procedure does not distinguishes among the stereo-isomers, i.e., molecules VII (1-Naphthol) and VIII (2-Naphthalelon); IX (2-Naphthalenamine) and X (1-Naphthalenamine) in Table 2, since furnishing the same values, respectively; instead the same quantum scheme is able to distinguish between the volumes of two stereo-isomers making the Equation (11) as a more general approach. This way, the molecular volumes are reported in the 6th column of Table 2 as computed within the ab initio–Hartree Fock method of Section 2.3.3.1; note that the HF method was chosen as the reference since it is at the “middle computational distance” between the semi-empirical and density functional methods; it has only the correlation correction missing; however, even the density functional schemes, although encompassing in principle correlation along the exchange–introduces approximations on the last quantum effect. Therefore, the molecular polarizability is computed upon the Equation (11) in the 7th column of Table 2 with the associate polarizability compactness aromaticities displayed in the 8th column of Table 2.

The molecular energetic reactivity indices of electronegativity and chemical hardness are computed upon the Equations (14) and (19) in terms of HOMO and LUMO energies computed within the quantum semi-empirical and ab initio methods presented in Section 2.3; their individual values as well as the resulted quantum compactness aromaticities, when combined with the AIM values of Table 2, in Equations (17) and (20) are systematically communicated in Tables 3 and 4, with adequate scaled representations in Figures 2 and 3, respectively.

Note that neither the minimal basis set (STO-3G) nor the single point computation frameworks, although both motivated in the present context in which only the bonding and the post-bonding information should be capped in computation, does not affect the foregoing discussion by two main reasons: (i) they have been equally applied for all molecules considered in all quantum methods’ combinations; and (ii) what is envisaged here is the aromaticity trend, i.e., the intra- and inter- scales comparisons rather than the most accurate values since no exact or experimental counterpart available for aromaticity.

Now, because of the observational quantum character of polarizability, one naturally assumes the (geometric) polarizability based- aromaticity scale of Table 2 as that furnishing the actual standard ordering among the considered molecules in accordance with the rule associated with Equation (12); it features the following newly introduced rules along possible generalizations

Under the reserve that these rules and their generalizations should be verified by extra studies upon a larger set of benzenoid aromatics, we will adopt them here in order to analyze their fulfillment with the energetically-based aromaticity scales of electronegativity and chemical hardness, reported in Tables 3 and 4 and drawn in Figures 2 and 3; actually, their behavior is analyzed against the aromaticity ordering rules given by Equations (17) and (20), i.e., as being anti-parallel and parallel with the polarizability-based aromaticity trend of Equation (12), with the results systematized in Tables 5 and 6, respectively.

From Table 5 there follows that electronegativity based-aromaticity displays the following properties respecting the aromaticity rules derived from polarizability framework:

No semi-empirical quantum method, in general, satisfies the first rule of aromaticity, Aroma1, in the sense that the trend in Figure 2a (and Table 3) displays rather growing aromaticity character from mono- to double-benzenoid rings; the same behavior is common also to HF computational environment, perhaps due to the close relationships with approximations made in semi-empirical approaches; instead, all other ab initio methods considered, including that without exchange and correlation terms in Equations (57), do fulfill the Aroma1 rule;

Overall, it seems electronegativity may be used in modeling compactness of atoms-in-molecules aromaticity–basically without counting on the exchange or correlation effects, or at best within the HF algorithm, while semi-empirical methods seems not well adequate. Yet, for all aromaticity rules formulated, there exists at least one quantum computational environment for which the electronegativity based compactness aromaticity is in agree with each of them.

The situation changes significantly when chemical hardness is considered for compactness aromaticity computation; the specific behavior is abstracted from the analysis of Table 6 and can be summarized as follows:

Semi-empirical methods are equally appropriate in producing agreement with Aroma1 and Aroma4 rules in what concerns the aromaticity behavior for the mono- versus bi- ring annulens and hydroxyl- versus amine- substitution to either of them, respectively;

Overall, when the chemical hardness agrees with one of the above enounced Aroma Rules it does that within more than one computational scheme; however, the best agreement of chemical hardness with polarizability based- aromaticity scales is for the mono- versus bi- (and possible poly-) benzenic rings decreasing of aromaticity orderings, along the manifestly hydroxyl- superior effects in aromaticity than amine- groups substitution within most of the computational quantum schemes, i.e., valid either for semi-empirical and ab initio methods. The stereoisomerism is not covered by chemical hardness modeling aromaticity, and along the electronegativity limited coverage within HF scheme in Table 5, there follows that the energetic reactive indices are not able to prevail over the geometric indices as polarizability or to predict stereoisomerism ordering in aromaticity modeling compactness schemes.

Finally, few words about the output of the various quantum computational schemes respecting the current aromaticity definition given by Equation (1) are worth addressing. As such, one finds that:

With CNDO and INDO methods, the electronegativity based-aromaticity is more oriented towards the AIM limit of Figure 1, while chemical hardness based- aromaticity merely models the MOL limit of chemical bonding, see Table 3. This agrees with the basic principles of chemical reactivity according to which electronegativity drives the atomic encountering in forming the transition state towards chemical bond, while chemical hardness refines the bond by the aid of maximum hardness principle [59,67];

The bottom line is rising by the wish to globally combine the ideas of quantum chemical methods used in chemical bonding, reactivity principles, and aromaticity results; upon the above discussions it follows that MINDO3, AM1 (or PM3)–for semi-empirical along Becke hybrid functionals and Hartree-Fock–for ab initio are the suited methods that fulfils most of the reactivity and the present introduced aromaticity bonding rules. However, the best of them overall seems to remain the consecrated HF scheme, since acquiring the highest number of grades summated throughout Tables 5 and 6. As such, a new challenge appears since the present results recommend that correlation does not count too much in aromaticity or reactivity modeling. Nevertheless, further studies with larger set of molecules and types of aromatics should be address for testing whether or not the advanced aromaticity (Aroma1–5) rules are preserved or in which degree they may be generalized or modified such that being in accordance with the principles of chemical bonding and reactivity.

Modeling the stability and reactivity of molecules is perhaps the greatest challenge in theoretical and computational chemistry. This is because the main conceptual tools developed as the reactivity indices of electronegativity and chemical hardness along the associate principles are often suspected by the lack of observability character. Therefore, although very useful in formal explanations of chemical bonding and reactivity, it is hard to find their experimental counterpart unless expressed by related measurable quantities as energy, polarizability, refractivity, etc. When the aromaticity concept come into play, it seems no further conceptual clarification is acquired, since no quantum observable or further precise definition can be advanced; in fact, the aromaticity concept associates either with geometrical, energetic, topologic, electronic molecular circuits (currents), or with the less favored entropic site in a molecule, just to name few of its representations.

However, since at the end, the aromaticity appears to describe the stability character of the molecular sample, its connection with a reactivity index seems natural, although systematically ignored so far. In this respect, the present work focuses on how the electronegativity and chemical hardness based-aromaticity scales behave with respect to others constructed on a direct observable quantum quantity–the polarizability in this case. This is because the polarizability quantity is fundamental in quantum mechanics and usually associated with the second order Stark effect that can be computed within the perturbation theory (see Appendix). Then, two ways of seeing a molecular structure were employed in introducing the actual absolute aromaticity definition:

the molecule viewed as composed of the constituent atoms (AIM) and

The two molecular perspectives may be associated with the pre- and post-bonding stages of a chemical bond at equilibrium; therefore, the conceptual and computational competition between these two molecular facets should measure the stability or its contrary effect - the reactivity propensity - being therefore the ideal ingredients for an absolute definition of aromaticity. Note that although an AIM-to-MOL difference definition of absolute aromaticity was recently advanced [38], the actual study of their ratio definition should account for a sort of compactness degree of molecular structure–as described by the specific molecular property used.

In short, for a molecular property to become a candidate for absolute (here with its compactness variant of) aromaticity, it has to fulfill two basic conditions:

having a viable quantum definition (since the quantum nature of electrons and nucleus are assumed as responsible for molecular stability/reactivity/aromaticity); and

In this respect, all the presently considered reactivity indices, i.e., polarizability, electronegativity, and chemical hardness, have equally consecrated quantum definitions as well as atomic and molecular representations [55,139].

At the atomic level, the experimental values based on the ionization potential and electron affinity definitions for electronegativity and chemical hardness were considered, see Equations (14) and (19), respectively, while for the polarizability new Hydrogenic quantum formulation was provided by Equation (6), and in Appendix by Equation (A22), recovering the exact value for the Hydrogen system by Equation (7), in close agreement with other available atomic quantum formulations of Equations (4) and (5). Nevertheless, the AIM level was formed by appropriate averaging of atoms-in-molecule summation for each of the considered reactivity indices, see Equations (8), (13) and (18), and along of their MOL counterparts of Equations (11), (14), and (19) the polarizability-, electronegativity- and chemical hardness- based aromaticity definitions were formulated with the associate qualitative trends established by Equations (12), (17), and (20), respectively. Yet, for MOL level of computations, all major quantum chemical methods for orbital spectra computation were considered, in Section 2.3, and implemented in the current application for some basics aromatics in Section 2. Because of the quantum observable character of polarizability the related aromaticity scale was considered as benchmark for actual study and it offered the possibility of formulating five rules for aromaticity:

Aroma1: the greater effect on aromaticity by mono- over bi-(poly-) benzenic rings;

These rules are then checked for electronegativity and chemical hardness derived-aromaticity scales with the synopsis of the results in Tables 5 and 6. It followed that chemical hardness, although generally in better agreement with these rules for most of the quantum chemical methods considered for its MOL computation, may not be considered infallible against aromaticity, at least for the reason it does not fulfils at all with the Aroms5 rule above. Surprisingly, chemical hardness index is more suited in modeling aromaticity when considered within semi-empirical computational framework, while the electronegativity responds better in conjunction with ab initio methods.

From quantum computational perspective, the consecrated HF method seems to get more marks in fulfillment of above Aroma1-to-5 rules, cumulated for electronegativity and chemical hardness based-aromaticity scales; it leads with the important idea the correlation effects are not determinant in aromaticity phenomenology, an idea confirmed also by the fact the density functional without exchange and correlation produces not-negligible fits with Aroma1, 2, and 4 rules in electronegativity framework.

Overall, few basic ideas in computing aromaticity should be finally emphasized

there is preferable computing aromaticity in an absolute manner, i.e., for each molecule based on its pre- and post- bonding properties (as is the present compactness definition, for instance) without involving other referential molecule, as is often case in the fashioned aromaticity scales;

Future quests should enlarge the basis of the present conclusions by performing comparative aromaticity studies at the level of biomolecules and nanostructures; at the end of the day, the aromaticity concept in general and with its particular specialization should represent just a tool/vehicle in modeling and understanding the chemical bond of atoms in molecules and nanostructures, either in isolated or interacting states.